We rely on the techniques and results illustrated in [1]}, [2]} and [3]}, as they provide a relatively easy way to identify which states are stochastically stable.[4]}, building upon [5]}, provides an asymmetry condition that implies stochastic stability and is applicable to a wide variety of settings.
More precisely, we restrict our attention to absorbing states (since there are no other recurrent states by virtue of Proposition REF ), and for any pair \((x,x^{\prime })\)  of absorbing states we define \(r^*(x,x^{\prime })\)  as the minimum sum of the resistances between states over any path starting in \(x\)  and ending in \(x^{\prime }\) .It is worth noting that here absorbing states refer to those of the corresponding unperturbed dynamics, \(\mathcal {A}\) , while no state is absorbing in the perturbed dynamics.
Then, for any absorbing state \(x\) , we define an \(x\) -tree as a tree having root at \(x\)  and all absorbing states as nodes. The resistance of an \(x\) -tree is defined as the sum of the \(r^*\)  resistances of its edges. Finally, the stochastic potential of \(x\)  is said to be the minimum resistance over all trees rooted at \(x\) .
